For x, y ≥ 1, let Ψ(x, y) denote the number of positive integers less than or equal to x and free of prime factors greater than y. The behaviour of the function Ψ(x, y) has been the object of numerous articles (see e.g. Norton's memoir [5] and the bibliography there). It turns out that a good approximation to ψ(x, y)/x is given by ρ(log x/log y), where the function ρ(t) is defined for t ≥ 0 as the continuous solution of the equations

Cardinal functions of topologies have been extensively studied. Cardinal functions of measures have attracted less interest, perhaps because there are fewer straightforward results which are independent of special axioms. In this paper I consider the “additivity” and “cofinality” of a measure (Definition 1) and show that they can often be calculated in terms of certain fundamental cardinals (Corollary 11 and Theorem 16).

Let 1 ≤ α ≤ β ≤ γ be cardinals, and let denote the class of all graphs on γ vertices having no subgraph isomorphic to Kα,β. A graph is called universal if every can be embedded into Go as a subgraph. We prove that, if α < ω ≤ γ and the General Continuum Hypothesis is assumed, then has a universal element, if, and only if, (i) γ > ω or (ii) γ = ω, α = 1 and β ≤ 3. Using the Axiom of Constructibility, we also show that there does not exist a universal graph in .

Gale transforms are constructed for certain infinite dimensional α-polytopes. In a manner analogous to the finite dimensional case the Gale transform can be used to determine all closed faces and Radon partitions of the α-polytope. A by-product is a characterization of closed faces using nets of functionals.

A recent paper [1] indicates that the beginnings of dynamic stall, near an aerofoil's leading edge, for instance, can be regarded as the finite-time nonlinear breakdown of a boundary layer subjected to an angle of attack above the critical value for the existence of a steady solution. The present theoretical study shows that the same non-linear breakdown can occur even in the below-critical regime. This happens particularly when reversed flow is present since short wavelength disturbances are then unstable and accumulate, for certain confined initial conditions, to force the finite-time collapse. A number of marginal cases with forward or reversed, subsonic or supersonic, oncoming motion are also noted and shed extra light on the instability and subsequent breakdown.

It is proved here that, if G is a positive definite integral ternary quadratic lattice of discriminant d and c is a squarefree integer which is primitively represented by the genus of G, then G primitively represents all sufficiently large integers of the type ct2, with g.c.d. (t, 2d) = 1, which are primitively represented by the spinor genus of G.

The principal objective of this work is to investigate various classes of centrally symmetric convex sets. These classes range from the zonoids at one extreme to the class of all centrally symmetric bodies at the other. The defining properties of these classes involve inequalities between mixed volumes. Various other characterizations will be found in response to a number of questions in a recent survey article by Rolf Schneider and Wolfgang Weil. Some of these are concerned with measures on a Grassmannian manifold while others relate to the intermediate surface area measures of convex bodies. We shall also show these classes are characterized by certain extremal geometric inequalities. The work concludes with a brief discussion of related results concerned with generalized zonoids.

We state some definitions belonging to the two halves of the title, going far enough to state our main results.

Fourier transforms. Let μ be a finite, complex-valued measure on R and its Fourier-Stieltjes transform. We define ℛ to be the set of μ with When μ ∈ ℛ and φ is of class (continuously differentiable of compact support), the identity shows that θ · μ ∈ ℛ.

Let K be a number field and E/K an elliptic curve. As is well known [3, 4,[ if K has class number 1, then there exists a global minimal Weierstrass equation for E. Our main goal in this paper is to prove the following converse to this statement.

In this paper the authors formulate a boundary-initial value problem for a linear elastic porous body saturated with an inviscid fluid and establish a continuous dependence theorem (Theorem 2) and two uniqueness theorems (Theorems 3, 4) for a particular class of such continua. Theorems 2, 3 are proved without hypotheses on the sign of the constants and, if the domain is unbounded, under mild assumptions on the spatial asymptotic behaviour of the field variables. Theorem 4 holds for body-forces not equal to zero and, if the domain is unbounded, without restrictions upon the behaviour of the unknown fields at infinity, but under suitable conditions on the sign of the constants.

In this note we present a simple way of obtaining the universal field of fractions of certain free rings as a subfield of an ultrapower of a (by no means unique) skew field. This method of embedding was discovered by Amitsur in [1]; our presentation uses Cohn's specialization lemma and the embedding is constructed in terms of full matrices over the rings in question (Theorem 3.2). In particular, if k is an infinite commutative field, the universal field of fractions of a free k-algebra can be realized as a subfield of an ultrapower of any skew extension of k, with centre k, which is infinite dimensional over k. Thus many problems concerning the universal field of fractions of a free k-algebra can be settled by studying skew extensions of k of relatively simple structure. More precisely and more generally, let E be a skew field with centre k and denote by R the free E-ring on X over k. Write U for the universal field of fractions of R and assume that U embeds in an ultrapower of a skew field D. Then by Łos' theorem, U inherits the first-order properties of D which can be expressed by universal sentences.

In 1942 Piccard [10] gave an example of a set of real numbers whose sum set has zero Lebesgue measure but whose difference set contains an interval. About thirty years later various authors (Connolly, Jackson, Williamson and Woodall) in a series of papers constructed F σ sets E in ℝ such that E – E contains an interval while the K-fold sum set

has zero Lebesgue measure for progressively larger values of k.

In this paper we are concerned with upper bounds for the sums

where

and Δ2(n) is written simply as Δ(n). These functions were introduced by Hooley [4] and applied in a novel way to problems related to Waring's, and in Diophantine approximation. Thus Hooley deduced from his result about S2(x) that for any irrational θ, real γ, and fixed ε > 0, the inequality

holds for infinitely many n. His result for S3(x) led to a proof that

where r8(n) denotes the number of representations of n as the sum of eight positive cubes.

Rather more than thirty years ago Erdős and Mirsky [2] asked whether there exist infinitely many integers n for which d(n) = d(n + 1). At one time it seemed that this might be as hard to resolve as the twin prime problem, see Vaughan [6] and Halberstam and Richert [3, pp. 268, 338]. The reasoning was roughly as follows. A natural way to arrange that d(n) = d(n + l) is to take n = 2p, where 2p + 1 = 3q, with p, q primes. However sieve methods yield only 2p + 1 = 3P2 (by the method of Chen [1]). To specify that P2 should be a prime q entails resolving the “parity problem” of sieve theory. Doing this would equally allow one to replace P2 by a prime in Chen's p + 2 = P2 result.

Let an be a non-increasing real sequence such that converges; then clearly an ↓ 0. We shall ignore the trivial case where an = 0 for all large n, and so we assume that an > 0 for all n, from now onwards. In [1] J. B. Wilker introduced certain new sequences associated with the rate of convergence of , and obtained various relations between them, in order to investigate packing problems in convex geometry. Let us define

We write P, Q and T respectively for the inferior limits of pn, qn and tn, and P1, Q1 and T1 for the corresponding superior limits. Further, we put

It is immediately clear from these definitions and our assumptions about an that

the latter since nan → 0 by Olivier's theorem [2, p. 124].

Let Φ : L → ℤ be a positive definite even unimodular quadratic form, L ≅ ℤn; we put f(x) = Φ(x)/2 and call (L, f) a lattice, for short. Let min f be the minimum of the numbers f(x) ≠ 0. Fixing n (a multiple of 8), one is interested in the largest possible minimum. It follows from the theory of modular forms (cf. Sloane [6]) that

In a recent paper, Parson and Sheingorn [11[ gave some estimates for certain exponential sums associated with Ramanujan's τ-function and, in general, with the Fourier coefficients of the cusp forms for the full modular group which are eigenfunctions for the Hecke operators. The exponential sums considered in that paper are closely connected with the exponential sum

where α ∈ ℝ and e(θ) = e2πiθ, and the methods used in [11] go back essentially to Hardy-Littlewood [5], Hardy [4], Hecke [6], Wilton [19] and Walfisz [18].

It has been shown in [1, 2] that it is sometimes possible to justify the uncoupled and quasi-static approximations which are commonly invoked to simplify the solution of initial and boundary value problems in the linear theory of thermoelasticity. The justification involves showing, among other things, that the temperature predicted by the coupled dynamic theory is approximated by a solution of the classical heat equation.